As in [1]}, all operations between the categories \(D_{\mathop {\mathrm {\acute{e}t}}}(-,\mathbf {Z}_\ell )\)  of \(\ell \) -adic complexes introduced above are always interpreted in the \(\ell \) -completed sense, i.e., the functor in question takes values in \(\ell \) -complete complexes by fiat, and agrees after reduction mod \(\ell \)  with the corresponding functor for finite coefficients. For instance, if \(j:U \rightarrow X\)  is a Zariski-open immersion and \(M \in D^b_{\mathop {\mathrm {\acute{e}t}}}(U,\mathbf {Z}_\ell ) \subset D^b_{\ell -\text{comp}}(U_v,\mathbf {Z}_\ell )\) , then \(j_! M \in D^b_{\ell -\text{comp}}(X_v,\mathbf {Z}_\ell )\)  is defined to be the derived \(\ell \) -completion of \(j^{top}_! M \in D^b(X_v,\mathbf {Z}_\ell )\)  where \(j^{top}_!\)  denotes the topos theoretic \(!\) -extension (without any completions); then one can see that \(j_! M\)  lies in \(D^b_{\mathop {\mathrm {\acute{e}t}}}(X,\mathbf {Z}_\ell )\)  and \(j_! M \otimes _{\mathbf {Z}_\ell } \mathbf {F}_\ell \)  agrees with \(j_!(M \otimes _{\mathbf {Z}_\ell } \mathbf {F}_\ell )\) , where the latter is defined in the classical way.

